A body moving with simple harmonic motion in a straight line has a velocity V and acceleration A, when the instaneous displacement X in cm from it maximum position is given by X is equal to 2.5sin0.4"t were t is in second, determine the magnitude of maximum velocity and determine the maximum velocity?
To determine the magnitude of maximum velocity and the maximum velocity of a body moving with simple harmonic motion, we'll need to analyze the given equation for displacement.
First, let's rewrite the equation for displacement:
X = 2.5 sin(0.4t)
We know that displacement is the integral of velocity, and velocity is the derivative of displacement with respect to time. Let's find the velocity function by taking the derivative of X with respect to t:
V = dX/dt = d/dt(2.5 sin(0.4t))
To differentiate sin(0.4t), we use the chain rule:
d/dt(sin(0.4t)) = 0.4 * cos(0.4t)
Now we can substitute this derivative back into the equation for velocity:
V = 2.5 * 0.4 cos(0.4t)
Next, we need to find the maximum velocity. To do this, we'll find the maximum value of the absolute value of V.
Since the cosine function oscillates between -1 and 1, the maximum value of cos(0.4t) is 1. Therefore, the maximum velocity occurs when cos(0.4t) = 1.
Substituting this into the equation for V:
V_max = 2.5 * 0.4 * 1
V_max = 1.0
Hence, the magnitude of the maximum velocity is 1.0 cm/s and the maximum velocity is 1.0 cm/s.
if x = A sin w t
then
v = dx/dt = A w cos w t
and
a = d^2x/dt^2 = -A w^2 sin w t = -w^2 x